The measurements shown earlier in Figs.  and  are raw (not processed or filtered), though it is common procedure to detrend data before estimating turbulence or associated return periods for a given turbulence level. Not all the extreme variance events are expected to be influenced by linear detrending, nor is such detrending necessarily appropriate for nonturbulent events; note, e.g., the event shown in Fig. c. Therefore we want to compare the 50-year return period of turbulence with the data detrended in two different ways: linear detrending and high-pass filtering. Detrending is performed by making a linear least-squares fit to the raw 10 min wind speed time series, with the linear component subsequently subtracted from the raw data.

The high-pass filtering is performed with a second-order Butterworth filter , whereby the magnitude of the frequency response function (the gain) is given by 2G(f)=11+(fc/f)4, where fc is the “cutoff” frequency. We perform the filtering using a cutoff frequency of 0.0017 Hz (1/600 Hz) and also with a higher cutoff frequency of 0.0033 Hz (1/300 Hz). The higher cutoff frequency chosen for the high-pass filtering corresponds to fluctuations with periods of 300 s (half of the period of the measurements). This choice of cutoff frequency ensures the removal of trends in the range 2.5–10 min (low-frequency transients) and is considered conservative enough to still include fluctuations associated with turbulent eddies.

Fluctuations with a period of 300 s at 4–25 m s-1 (the operational wind speed range of a typical wind turbine) correspond to length scales of 1200–7500 m. Length scales in this range are significantly larger than turbulent length scales that have been estimated at the Høvsøre site e.g.,.

Here we use the inverse first-order reliability method (IFORM) to estimate the 50-year return period contour corresponding to the joint description of turbulence (σu) and 10 min mean wind speed (U). This method was developed by and provides a practical way to evaluate joint extreme environmental conditions at a site. The IFORM method is widely used in wind energy to predict extreme environmental conditions or long-term loading on wind turbines for ultimate strength analysis. More information on this method may be found in, e.g., , , and .

The first step in the IFORM analysis is to find the joint probability distribution f(U,σu). According to the IEC standard the 10 min mean wind speed is assumed to follow a Weibull distribution

Here we use a three-parameter Weibull distribution. This is done because after filtering out measurements with errors and missing periods, the lowest mean wind speed is 2.2 m s-1. One could also use a weighted two-parameter Weibull distribution fit with increased weights in the tail to obtain the same result.

In Fig. , μσu and σσu are shown as functions of 10 min mean wind speed from Høvsøre unprocessed measurements at 100 m (grey dots) and the expressions from the IEC standard (blue lines) with Iref=0.12. The green lines show a third- and a second-order polynomial fit to the binned measurements of μσu and σσu, respectively (bins of 1 m s-1). The IEC expression for μσu is higher than that from the measurements but has a similar slope for mean wind speeds above 15 m s-1. The difference is larger between the data and IEC expression for σσu, for which the assumption of no mean wind speed dependency does not fit well to the data.

The next step in the IFORM analysis is to obtain a utility “reliability index” β, which translates the desired return period Tr (here 50 years) into a normalized measure corresponding to the number of standard deviations of a standard Gaussian distribution: 5β=Φ-11-TtTr=Φ-11-15nm. Here Φ-1 is the inverse Gaussian cumulative distribution function (CDF), Tt is the duration of a turbulence measurement (here 10 min), and nm is the number of 10 min measurements corresponding to a 10-year period (which equals the time span of the data). Thus, the reliability index equals the radius of a circular contour in standard Gaussian space so that 6β=u12+u22, where the standard normal variables u1 and u2 are derived from physical variables using an iso-probabilistic transformation, which takes correlations into account. We invoke the Rosenblatt transformation , which relies on the fact that a multivariate distribution may be expressed as a product of conditional distributions: F(x1,x2)=F(x1)F(x2|x1). In this analysis, only two variables are considered, and the transformation may be performed in the following way: 7U=FU-1Φ(u1),σu=Fσu|U-1Φ(u2), where FU is the three-parameter Weibull CDF and Fσu|U is the conditional lognormal CDF.

Figure  shows the joint distribution of mean wind speed and turbulence

Note that some measurement points have been removed due to measurement errors; therefore, the points are fewer than in Fig. , which includes 10 min statistics from the whole measurement period.

The peak and the corresponding location of each event are identified in the following way: a moving average is subtracted from the wind speed signal and the maximum value of the differences identified: 8upeak=max⁡(u-u‾60s), where u is the horizontal wind speed signal and u‾60s is the moving average over 60 s. The peaks are not necessarily the highest value of the signal, but rather the highest value within a sharp wind speed increase.

Applying the selection criteria described in Sect.  results in 99 identified events. Of these, 30 events are discarded as they include periods of missing measurements. A lower threshold of 4 m s-1 is put on upeak to exclude events mostly consisting of a linear trend or relatively insignificant peaks. Finally, events during which the corresponding directional data fluctuated below 180∘ are discarded, i.e., temporary directional data from the south, to exclude measurements from the wake of the nearby wind turbine. A remaining 44 events are chosen for load simulations. The measured time series including the extreme events are used to generate constrained turbulence simulations (explained in more detail in Sect. ) of 600 s duration. The time series period is selected such that the sharp wind increase, or ramp, occurs approximately in the middle of the time series, i.e., approximately 300 s before and after the peak.

Wind turbine response in the time domain is calculated with HAWC2 Horizontal Axis Wind turbine simulation Code 2nd generation;. HAWC2 is based on a multibody formulation for the structural part, whereby each body consists of Timoshenko beam elements. All the main components of a wind turbine are represented by these independent bodies and connected with different kinds of algebraic constraints. The aerodynamic forces are accounted for with blade element momentum theory see, e.g., with additional correction models: a tip correction model, a skewed inflow correction, and a dynamic inflow correction. HAWC2 additionally includes models that account for dynamic stall, wind shear effects on induction, tower-induced drag, and tower shadow.

All the load simulations are performed using the DTU 10 MW reference wind turbine (RWT), which is a virtual wind turbine model based on state-of-the-art wind turbine design methodology. The main characteristics of the RWT may be seen in Table , and a more detailed description may be found in . The controller used for the RWT is the Basic DTU Wind Energy controller .

The Mann spectral turbulence model is fully integrated into HAWC2, whereby a turbulence “box” may be generated for every wind turbine response simulation. The turbulence box is a three-dimensional grid that contains a wind velocity vector at each grid point. The turbulence boxes in this study all have 8192×32×32 grid points in the x, y, and z directions, respectively. The y–z plane is parallel to the rotor, and the distance between the grid points is typically defined so that the domain extent in the y and z directions becomes a few percent larger than the rotor diameter. The length of the x axis (Lx) is proportional to the mean wind speed at hub height, Lx=U⋅T, where T is the simulation time. The turbulence box is transported with the average wind speed at hub height through the wind turbine rotor.

The Mann model is based on an isotropic von Kármán turbulence spectral tensor, which is distorted by vertical shear caused by surface friction. Assumptions of constant shear and neutral atmospheric conditions in the rapid distortion limit are used to linearize the Navier–Stokes equations, which may then be solved as simple linear differential equations. The solution results in a spectral tensor that may be used in a Fourier simulation to generate a random field with anisotropic turbulent flow. The Mann model contains three parameters, as described below.

Γ is an anisotropy parameter; when positive, σu2>σv2>σw2, which are the variances of the u, v, and w components of the wind speed, respectively. When Γ=0, the generated turbulence is isotropic, σu2=σv2=σw2.

The IEC-recommended values of the parameters are Γ=3.9 and L=29.4 m (for hub heights above 60 m), and αε2/3 is set to a positive value to be scaled with σu2. It has been shown in numerous studies that these parameters can change significantly, e.g., with turbulence level , atmospheric stability , and site conditions . As we do not want to investigate the effect of changing these parameters, all turbulence realizations are chosen to have the same parameters. In the present study, the anisotropy parameter is chosen according to the IEC standard, Γ=3.9. The turbulence length scale is chosen differently because the DTU 10 MW RWT is a relatively large wind turbine, and the turbulence length scale is expected to be of the same order of magnitude as the hub height . Here the length scale is estimated via 9L=σudU/dz, as derived by . The final 200 s of simulation data, i.e., after the wind speed ramps, are used to estimate the length scale of turbulence and thus exclude the large coherent structure. Here σu from 100 m of height is used, along with dU/dz estimated between z=160 m and z=60 m. Using Eq. () the length scale is found on average to be 〈L〉≈120 m over all events analyzed. The value chosen is therefore L=120 m.

The DLC is simulated based on the setup described in , wherein mean wind speeds at hub height of 4–26 m s-1 in steps (bins) of 2 m s-1 are used, and each simulation has a duration of 600 s

In contrast with , here the simulations are performed without yaw misalignment.

The aim here is to generate turbulence simulations resembling the measured wind field of the extreme variance events. This is done by constraining the synthesized turbulence fields. The constraining procedure involves modifying the time series to represent the most likely realization of a random Gaussian field that would satisfy the constraints using an algorithm described in and demonstrated with applications to wind energy in and . For the constraining procedure we define three different random Gaussian fields as a function of location, r={x,y,z}:

the constrained field, f(r), which is the generated field of the procedure, modified to resemble the measurements;

A realization of the constrained field is generated by adding the conditional ensemble mean of the residual field to the source field: 12f(r)=f̃(r)+〈g(r)|C〉.

Here the constraints consist of the u and v components of the wind velocity measurements from the light mast. The constraints are applied at three different heights: 79 m, 119 m (hub height), and 179 m, i.e., shifted up 19 m so the measurements at 100 m represent hub-height wind speed. The constraints are also applied at three different widths (along the y axis): 89.6 m (the middle of the turbulence box) ±70 m. This is done to ensure the coherent structure of the observed flow in the simulations. Every third measurement is applied at each width along the y axis, giving applied constraints at each y location with a 3.33 Hz frequency. This is done to reduce the number of applied constraints and thereby the computational time of the simulations.

In Fig.  two turbulence boxes with different random seeds are seen. The u component of the turbulent field is shown with a color scale on slices along the time axis. Figure a–b show the unconstrained turbulence boxes, and Fig. c–d show the same turbulence boxes with constraints corresponding to measurements from two different extreme variance events.

Figure  shows two examples of the u velocity time series at hub height with and without applied constraints for the same turbulence seeds as shown in Fig. .

For the purpose of load simulations, six different constrained turbulence seeds are generated from each extreme variance event time series. Although applying the constraints makes the turbulence boxes similar in general, there are differences in the parts of the boxes that are far from the constraint locations. As a result, there will be a seed-to-seed variation in loads simulated with constrained turbulence boxes, but they are much smaller than what is seen in the unconstrained case.

In Fig.  the standard deviation of the simulated hub-height u component wind speed is shown as a function of the mean hub-height u component wind speed. Each dot shows the standard deviation averaged over six turbulence seeds. As the variance is scaled to match the target for both DLC 1.3 and the constrained simulations, the scatter of the mean standard deviation over the six different seeds is small. The standard error of the mean standard deviation is in the range of 0.008–0.013 m s-1, and the standard error of the mean hub-height u component wind speed is equal to or less than 0.015 m s-1. The standard deviation from the constrained turbulence simulations (blue dots) is higher than that of DLC 1.3 with one exception. For this case, some variance was lost as a consequence of changing the time interval selection to span ±300 s around the wind speed peak, and data with a negative trend were cut off.

In Fig.  the characteristic extreme loads from DLC 1.3 and the constrained simulations are compared. The maximum–minimum load values of each 10 min HAWC2 simulation are binned according to wind speed with a bin width of 2 m s-1 and then averaged. For the comparison we omit the wind speed bin at 26 m s-1, as there are no observed events within that wind speed bin. The error bars show the standard deviation of the extreme loads of each wind speed bin. Both maxima and minima are shown for the tower-top moments, but for all other load components only the maximum moments are shown. It should be noted that the in-plane blade root flap moment maxima are negative due to the orientation of the blade coordinate system of the wind turbine model in HAWC2.

Figure a and b show the extremes of the tower-top tilt and yaw moments, respectively. In the whole wind speed range the mean extreme moments for DLC 1.3 are between 6400 and 21 000 kNm larger than for the constrained simulations.

Figure c shows the mean extreme tower-base fore–aft moments. The overall highest mean extreme moment is from the DLC 1.3 simulation set; however, for the constrained turbulence simulations the loads are higher for wind speed bins at 8 m s-1 and between 14 and 20 m s-1. The largest difference is seen for wind speed bin 16 m s-1, wherein the mean extreme moment from the constrained simulation is 50 200 kNm larger than from the DLC 1.3.

Figure d shows the mean extreme tower-base side–side moments. In the whole wind speed range the mean extreme moments for the DLC 1.3 are between 6000 and 22 500 kNm larger than for the constrained simulations.

Figure e and f show the blade root flap and edge moments, respectively. In the whole wind speed range the mean extreme moments for the DLC 1.3 are between 800 and 6200 kNm larger than for the constrained simulations, with the exception of wind speed bin 16 m s-1, wherein the mean extreme moments from the constrained simulations are respectively 3000 and 400 kNm higher than the DLC 1.3.

The extreme tower-top tilt, yaw, and tower-base side–side moments show a general increase with wind speed. The extreme blade root flap and tower-base fore–aft moments peak around rated wind speed. For the extreme blade root edge moment it is seen that the loads peak around rated wind speed for both simulation sets, but the main difference is that after 16 m s-1 the DLC 1.3 loads and the scatter increase with wind speed.

Table  lists the overall characteristic loads from each simulation set (the extremes seen in Fig. ), together with their ratio. The difference between the overall extremes from the two simulation sets is largest for the tower-top yaw moment, wherein the extremes are lower from the constrained simulations. The overall extremes are of similar magnitude for the tower-base fore–aft moment and the blade root flap-wise moment.

In the following, examples of 10 min time series from DLC 1.3 and constrained simulation sets are shown side by side for comparison and a demonstration of the differences in the wind turbine response to different types of wind regime. A comparison is made for the tower-base fore–aft moment, wherein the characteristic extreme loads from the different simulation sets are of similar magnitude. We also consider and compare the tower-top tilt and yaw moments, which give the largest differences between the two simulation sets.

First, we compare two time series giving some of the highest extreme tower-base fore–aft moments from each simulation set. For DLC 1.3 in Fig.  the mean u component hub-height wind speed is U=12.0 m s-1, with a standard deviation of σu=2.7 m s-1, and the peak tower-base fore–aft moment is 236 000 kNm. For the constrained simulation, U=14.9 m s-1 and σu=3.5 m s-1. The peak tower-base fore–aft moment is 228 000 kNm. The peak tower-base fore–aft moments are of similar magnitude in the simulations, and in both cases this occurs when the pitch angle is zero degrees – right before the wind turbine blades begin to pitch. Also, at the time when the wind speed at hub height reaches rated wind speed, the wind speed at 179 m is above rated wind speed, leading to higher loading on the upper half of the rotor. From the turbulence simulations, the most noticeable difference in the wind turbine response is that in the constrained turbulence simulation the time of the peak tower-base fore–aft moment is very distinguishable at 390 s. While for the stationary turbulence the peak response occurs around 150 s, numerous times it reaches above 200 000 kNm during the simulation. Note that the axes in Fig. a and b are the same, as are the axes in Fig. c and d. It is seen that although the standard deviation of the wind speed is lower in the stationary turbulence simulation, the wind speed extremes are greater, with instantaneous wind speed reaching below 2 m s-1 and above 22 m s-1.

In Fig.  we compare some of the most extreme tower-top moments from the two simulation sets. The stationary turbulence simulation in Fig.  has U=22 m s-1 and σu=3.4 m s-1, with a peak tower-top tilt moment of 36 601 kNm and a peak tower-top yaw moment of -28 900 kNm; in contrast, the constrained turbulence simulation has U=21.3 m s-1 and σu=6.6 m s-1, with a peak tower-top tilt moment of 30800 kNm and a peak tower-top yaw moment of -18 600 kNm. As in the previous example, the time of peak loads is very clearly identified in the constrained turbulence simulation, and the peak value is significantly higher than the response for the remainder of the simulation. For the stationary turbulence simulation, the tower-top yaw and tilt moments often reach high values throughout the simulation. Extreme tower-top moments tend to be observed when there is high shear across the rotor. In stationary turbulent flow the variation in wind speed across the rotor arises as turbulent eddies sweep by, hitting only part of the rotor, leading to high wind shear. The extreme tower-top loads from the constrained simulations are in connection with high vertical wind shear arising during the wind speed increase (ramp event).